Twelvefold Way. Inclusion-Exclusion-Principle & Möbius Inversion. Newton s Binomial Theorem
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1 1 Permutations and Combinations Multinomial Coefficients Twelvefold Way Cycle Decompositions 2 Inclusion-Exclusion-Principle & Möbius Inversion PIE Möbius Inversion Formula 3 Generating Functions Ordinary and Exponential Newton s Binomial Theorem Reccurence Relations
2 4 Partitions (Non-Crossing) Partitions of [n] Ferrer Diagrams Standard Young Tableaux 5 Partially Ordered Sets (Symmetric) Chain Partitions Dimension Posets Between Two Lines 6 Designs Existence/ Non-Existence Steiner Triple Systems etc. Latin Squares
3 1 Permutations and Combinations Multinomial Coefficients Twelvefold Way Cycle Decompositions ( n r 1,...,r k ) = n! r 1!... r k! where n = r r k # n-permutations of multiset M = {r 1 t 1,..., r k t k } Multinomial Theorem: ( ) (x x k ) n n = r 1,...,r k x r 1 1 x r k k r r k = n
4 1 Permutations and Combinations Multinomial Coefficients Twelvefold Way Cycle Decompositions n balls k boxes 1 per box 1 per box arbitrary ( U L k ( n 1 ) ( n+k 1 ) n) k 1 k 1 L U 1 s II k ) (n) L L n! s II k (n)k! ( k n U U 1 p k (n) k i=1 sii i kn (n) k i=1 p i(n)
5 1 Permutations and Combinations Multinomial Coefficients Twelvefold Way Cycle Decompositions π = (123)(5)(46) = = unique partition into disjoint cycles exactly k cycles s I k (n) permutations no trivial cycles derangements
6 2 Inclusion-Exclusion-Principle & Möbius Inversion PIE Möbius Inversion Formula properties P 1,..., P m N(S) = {x x has P i for all i S} S [m] Then S [m] ( 1) S N(S) elements have none of the properties. Usage: Define P i Bound N(S) Apply PIE
7 2 Inclusion-Exclusion-Principle & Möbius Inversion PIE Möbius Inversion Formula Stronger PIE g(a) = f(s) f(a) = ( 1) A S g(s) S A S A Möbius Inversion g(n) = f(d) f(n) = µ(d)g( n d ) d n d n General Posets g(y) = f(x) f(y) = µ(x, y)g(x) x y x y
8 3 Generating Functions Ordinary and Exponential Newton s Binomial Theorem Reccurence Relations F (x) = f n x n n 0 A(x) B(x) = ( a k b n k ) x n n 0 k = 0 n unlabeled objects G(x) = g n x n n! n 0 A(x) B(x) = n ( ( n ) k) ak b xn n k n! n 0 k = 0 labeled objects
9 3 Generating Functions Ordinary and Exponential Newton s Binomial Theorem Reccurence Relations Newton s Binomial Theorem (1 + x) n = ( n k) x k n k = 0 n R {0} F (x) = 1 + x F (x) 2 F (x) = 1 n 0 Catalan Numbers: C n = 1 n+1( 2n n n+1( 2n n ) ) x n
10 3 Generating Functions Ordinary and Exponential Newton s Binomial Theorem Reccurence Relations reccurence p(a)f = g initial values f(0) = f 0,..., f(k) = f k homogeneous (g = 0): p(a) = (A r) k f(n) = c 1 r n + + n k 1 c k r n p(a) = p 1 (A) p 2 (A) f(n) = f 1 (n) + f 2 (n) non-homogeneous (g 0): f 0 (n) gen. sol. f 1 (n) particular sol. of hom. system of non-hom. system f = f 0 + f 1
11 4 Partitions (Non-Crossing) Partitions of [n] Ferrer Diagrams Standard Young Tableaux Bell Numbers n B n = s II k (n) k = 0 # ways to split n persons in groups Non-Crossing Partitions counted by Catalan numbers C n = 1 n+1( 2n n )
12 4 Partitions (Non-Crossing) Partitions of [n] Ferrer Diagrams Standard Young Tableaux p(n) = # ways to write n as a sum Pentagonal Numbers ω k p even d (n) p odd d (n) 11 = = Thm. p odd (n) = p dist (n) ( 1) k if n = ω k 0 otherwise
13 4 Partitions (Non-Crossing) Partitions of [n] Ferrer Diagrams Standard Young Tableaux π = RSC T 1 5 T 2 Hook Length Formula 7 t(λ) = n! hi,j (i, j)
14 5 Partially Ordered Sets (Symmetric) Chain Partitions Dimension Posets Between Two Lines Dilworth s Theorem Multiset Lattices partition into w(p ) chains symmetric chain partition
15 5 Partially Ordered Sets (Symmetric) Chain Partitions Dimension Posets Between Two Lines dim(p ) = min{k P = L 1 L k } Thm: dim(p ) w(p ) dim(s n ) = w(s n ) = n Thm: dim(p ) 2 L ordering all 1 2
16 5 Partially Ordered Sets (Symmetric) Chain Partitions Dimension Posets Between Two Lines interval order no 2 2 segment order dim(p ) 2 triangle order curve order L ordering all 2 2 all posets
17 6 Designs Existence/ Non-Existence Steiner Triple Systems etc. Latin Squares t-(v, k, λ) Design: v points, blocks are k-sets, every t-tuple of points in λ blocks Thm: # blocks = λ (v ( t) / k ) t repetition of i-sets = λ (v i ) ( / k i ) t i t i Thm: # blocks v (t + 1)(k t + 1) 2-(7, 3, 1)-design
18 6 Designs Existence/ Non-Existence Steiner Triple Systems etc. Latin Squares Affine Planes 2-(n 2, n, 1)-designs n prime power Steiner Triple Systems 2-(v, 3, 1)-designs v {1, 3} (mod 6) Projective Planes 2-(q 2 + q + 1, q + 1, 1)-designs q prime power
19 6 Designs Existence/ Non-Existence Steiner Triple Systems etc. Latin Squares Thm n 1 MOLS of order n n-by-n array filled with Z n each row is a permutation each column is a permutation affine plane of order n 2 n = p k
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